Advanced search
1 file | 611.01 KB

Cubic vertices in planar hypohamiltonian graphs

(2019) Journal of Graph Theory. 90(2). p.189-207
Author
Organization
Abstract
Thomassen showed in 1978 that every planar hypohamiltonian graph contains a cubic vertex. Equivalently, a planar graph with minimum degree at least 4 in which every vertex‐deleted subgraph is hamiltonian, must be itself hamiltonian. By applying work of Brinkmann and the author, we extend this result in three directions. We prove that (i) every planar hypohamiltonian graph contains at least four cubic vertices, (ii) every planar almost hypohamiltonian graph contains a cubic vertex, which is not the exceptional vertex (solving a problem of the author raised in J. Graph Theory [79 (2015) 63–81]), and (iii) every hypohamiltonian graph with crossing number 1 contains a cubic vertex. Furthermore, we settle a recent question of Thomassen by proving that asymptotically the ratio of the minimum number of cubic vertices to the order of a planar hypohamiltonian graph vanishes.
Keywords
Combinatorics, Graph Theory, Hamiltonian

Downloads

  • Zamfirescu-2019-Journal of Graph Theory.pdf
    • full text
    • |
    • open access
    • |
    • PDF
    • |
    • 611.01 KB

Citation

Please use this url to cite or link to this publication:

Chicago
Zamfirescu, Carol. 2019. “Cubic Vertices in Planar Hypohamiltonian Graphs.” Journal of Graph Theory 90 (2): 189–207.
APA
Zamfirescu, C. (2019). Cubic vertices in planar hypohamiltonian graphs. Journal of Graph Theory, 90(2), 189–207.
Vancouver
1.
Zamfirescu C. Cubic vertices in planar hypohamiltonian graphs. Journal of Graph Theory. Wiley; 2019;90(2):189–207.
MLA
Zamfirescu, Carol. “Cubic Vertices in Planar Hypohamiltonian Graphs.” Journal of Graph Theory 90.2 (2019): 189–207. Print.
@article{8588698,
  abstract     = {Thomassen showed in 1978 that every planar hypohamiltonian graph contains a cubic vertex. Equivalently, a planar graph with minimum degree at least 4 in which every vertex\unmatched{2010}deleted subgraph is hamiltonian, must be itself hamiltonian. By applying work of Brinkmann and the author, we extend this result in three directions. We prove that (i) every planar hypohamiltonian graph contains at least four cubic vertices, (ii) every planar almost hypohamiltonian graph contains a cubic vertex, which is not the exceptional vertex (solving a problem of the author raised in J. Graph Theory [79 (2015) 63--81]), and (iii) every hypohamiltonian graph with crossing number 1 contains a cubic vertex. Furthermore, we settle a recent question of Thomassen by proving that asymptotically the ratio of the minimum number of cubic vertices to the order of a planar hypohamiltonian graph vanishes.},
  author       = {Zamfirescu, Carol},
  issn         = {0364-9024},
  journal      = {Journal of Graph Theory},
  language     = {eng},
  number       = {2},
  pages        = {189--207},
  publisher    = {Wiley},
  title        = {Cubic vertices in planar hypohamiltonian graphs},
  url          = {http://dx.doi.org/10.1002/jgt.22388},
  volume       = {90},
  year         = {2019},
}

Altmetric
View in Altmetric